Power-law sensitivity to initial conditions—New entropic representation

The exponential sensitivity to the initial conditions of chaotic systems (e.g. D = 1) is characterized by the Liapounov exponent λ, which is, for a large class of systems, known to equal the Kolmogorov-Sinai entropy K. We unify this type of sensitivity with a weaker, herein exhibited, power-law one through (for a dynamical variable x) (equal to eλ1t for q = 1, and proportional, for large t, to for q ≠ 1;. We show that gl ( q), where Kq is the generalization of K within the non-extensive thermostatistics based upon the generalized entropic form . The well-known theorem λ1 = K1 (Pesin equality) is thus extended to arbitrary q. We discuss the logistic map at its threshold to chaos, at period doubling bifurcations and at tangent bifurcations, and find , respectively. 05.45. + b; 05.20. − y; 05.90. + m.